Study guide
11.4 Circles in the Coordinate Plane
ACCUPLACER Advanced Algebra and Functions Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Scoring, Skills Insight Bands, and Placement 1.3 How to Study for AAF
2 Linear Equations
2.1 Writing Linear Equations in One and Two Variables 2.2 Solving Linear Equations 2.3 Linear Inequalities 2.4 Systems of Two Linear Equations
3 Linear Applications and Graphs
3.1 Slope, Intercepts, and Graphing Lines 3.2 Parallel and Perpendicular Lines 3.3 Modeling with Linear Functions 3.4 Graphing Linear Inequalities and Systems
4 Factoring
4.1 GCF, Grouping, and Special Products 4.2 Factoring Quadratic Trinomials 4.3 Factoring Cubics and Higher Polynomials
5 Quadratics
5.1 Quadratic Forms, Graphs, Vertex, and Intercepts 5.2 Solving Quadratics by Factoring 5.3 Quadratic Formula and Completing the Square 5.4 Quadratic Inequalities and Quadratic-Linear Systems
6 Functions
6.1 Function Notation, Evaluation, Domain, and Range 6.2 Graphs of Functions, Vertical Line Test, Maxima and Minima 6.3 Function Operations, Composition, and Inverses 6.4 Transformations and Interpreting Functions in Context
7 Radical and Rational Equations
7.1 Simplifying Radical Expressions 7.2 Solving Radical Equations and Extraneous Roots 7.3 Rational Expressions and Equations 7.4 Graphs, Domain, and Range of Radical and Rational Functions
8 Polynomial Equations
8.1 Polynomial Operations and Structure 8.2 Solving Polynomial Equations 8.3 Graphing Polynomials, Roots, and End Behavior
9 Exponential and Logarithmic Equations
9.1 Exponential Functions, Growth, and Decay 9.2 Graphs of Exponential Functions 9.3 Logarithms and Logarithmic Properties 9.4 Solving Exponential and Logarithmic Equations
10 Geometry Concepts for Algebra 1
10.1 Area, Perimeter, and Prism Volume 10.2 Distance Formula and Pythagorean Theorem 10.3 Dilations, Rotations, Translations, and Reflections
11 Geometry Concepts for Algebra 2
11.1 Volume of Cylinders, Cones, and Spheres 11.2 Intersecting Lines and Angle Relationships 11.3 Triangle Similarity and Congruence 11.4 Circles in the Coordinate Plane
12 Trigonometry
12.1 Right-Triangle Trigonometry and Special Triangles 12.2 Unit Circle, Radians, and Arc Length 12.3 Trigonometric Graphs and Identities 12.4 Solving Trig Equations; Law of Sines and Cosines
11.4 Circles in the Coordinate Plane
Key Takeaways Explain This Topic AI
Standard form is (x − h)^2 + (y − k)^2 = r^2 with center (h, k) and radius r. (x + 1)^2 means the center x-coordinate is −1, not +1.
Completing the square on x^2 + y^2 − 6x + 4y − 3 = 0 gives (x − 3)^2 + (y + 2)^2 = 16, so center (3, −2) and radius 4.
Compare (x − h)^2 + (y − k)^2 to r^2 to classify a point: less than r^2 is inside, equal is on the circle, greater is outside.
On (x − 3)^2 + (y + 2)^2 = 16, point (6, 2) has squared distance 25 > 16 so it is outside; (3, 2) has squared distance 16 so it is on the circle.
Radius is the square root of the right-hand side, never r^2 itself and never negative. Add the completed-square constants to both sides.
Last updated: August 2026
11.4 Circles in the Coordinate Plane
College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Geometry concepts for Algebra 2 at 5–10% , typically 1–2 CAT items . Table 11’s last Algebra 2 geometry skill is circle equations in the coordinate plane . The standard form is
(x − h)^2 + (y − k)^2 = r^2
The center is (h, k) and the radius is r (positive). This is the Pythagorean theorem / distance formula: every point (x, y) on the circle is exactly r units from the center. Distance formula from Algebra 1 geometry is in Distance Formula and Pythagorean Theorem . Completing the square is the same algebra as vertex form for a quadratic, now done once in x and once in y — see Quadratic Formula and Completing the Square if that move is rusty. Work FREE mixed AAF items at /practice/accuplacer-advanced-algebra after you can convert general form to standard form without dropping a constant.
- Reading standard form
- (x − 3)^2 + (y + 2)^2 = 16
Center (3, −2) , because y + 2 = y − (−2) , and r^2 = 16 so r = 4 . Radius is never negative. r^2 = 16 does not make r = −4 .
- Equation Center Radius
- (x − 3)^2 + (y + 2)^2 = 16 (3, −2) 4
- (x + 1)^2 + (y − 5)^2 = 9 (−1, 5) 3
- x^2 + y^2 = 25 (0, 0) 5
- (x − 4)^2 + y^2 = 7 (4, 0) √7
Trap: flipping the sign of h or k . (x + 1)^2 is center x-coordinate −1 , not +1. Trap: reporting r = 16 or r = 9 (forgetting to square-root). Trap: writing the right-hand side as the radius when the stem asks for r .
Worked: write the equation from a center and a point
Center (2, −1) , circle passes through (5, 3) .
Radius is the distance: r^2 = (5 − 2)^2 + (3 − (−1))^2 = 9 + 16 = 25 .
Equation: (x − 2)^2 + (y + 1)^2 = 25 .
You can leave r as 5, but the equation wants r^2 on the right side. Do not write = 5 on the right; that would be r , not r^2 .
Worked: diameter endpoints
A circle has diameter from A(−1, 4) to B(5, −2) . The center is the midpoint:
h = (−1 + 5)/2 = 2 , k = (4 + (−2))/2 = 1 , so center (2, 1) .
r^2 is the squared distance from the center to A (or half the diameter, squared):
(−1 − 2)^2 + (4 − 1)^2 = 9 + 9 = 18
Equation: (x − 2)^2 + (y − 1)^2 = 18 . Radius is √18 = 3√2 , but leaving r^2 = 18 in the equation is the standard-form requirement.
- Completing the square: the AAF workhorse
- General form: x^2 + y^2 + Dx + Ey + F = 0
Group x terms, group y terms, move the constant, complete both squares.
- Worked: x^2 + y^2 − 6x + 4y − 3 = 0
- Group: (x^2 − 6x) + (y^2 + 4y) = 3
Complete x: take half of −6, which is −3, square to 9. Add 9.
Complete y: take half of +4, which is +2, square to 4. Add 4.
Add 9 and 4 to both sides:
- (x^2 − 6x + 9) + (y^2 + 4y + 4) = 3 + 9 + 4
- (x − 3)^2 + (y + 2)^2 = 16
Center (3, −2) , radius 4 .
The linear x-coefficient is −6, so h = −D/2 = 6/2 = 3 if you write general form as x^2 + y^2 + Dx + Ey + F = 0 with D = −6 . The linear y-coefficient is +4, so k = −E/2 = −2 . Same result as completing the square; use whichever you can do without dropping a sign.
Trap: forgetting to add 9 and 4 to the right side, leaving = 3 and claiming r = √3 . Trap: (y − 2)^2 from the +4y term — half of +4 is +2, so (y + 2)^2 , center y-coordinate −2 . Trap: radius 16. Trap: moving −3 to the right as −3 instead of +3.
Walk the constant one more time, because that is where most CAT misses happen. Starting from x^2 + y^2 − 6x + 4y − 3 = 0 , add 3 to both sides first: x^2 − 6x + y^2 + 4y = 3 . Then add 9 and 4. The right side is 3 + 9 + 4 = 16 , not −3 + 9 + 4 . The original −3 is a left-side constant that becomes +3 on the right before the completed-square constants join it.
- Another worked: x^2 + y^2 + 8x − 2y + 13 = 0
- (x^2 + 8x) + (y^2 − 2y) = −13
Half of 8 is 4, square 16. Half of −2 is −1, square 1.
(x + 4)^2 + (y − 1)^2 = −13 + 16 + 1 = 4
Center (−4, 1) , radius 2.
If the right side came out negative , there is no real circle (empty set). If it came out zero , the graph is a single point (degenerate circle of radius 0). AAF multiple-choice usually gives a genuine circle.
- Worked: a missing y-linear term
- x^2 + y^2 − 10x + 21 = 0
- (x^2 − 10x) + y^2 = −21
Complete only x: half of −10 is −5, square 25.
(x − 5)^2 + y^2 = −21 + 25 = 4
Center (5, 0) , radius 2. When a linear term is missing, that coordinate of the center is 0. Do not invent a (y − 21) shift from the constant term.
Inside, on, or outside
Compare the distance from the center to a point with the radius.
- Distance < r : inside
- Distance = r : on the circle
- Distance > r : outside
You can compare squared distances to r^2 to avoid a square root. That is the AAF-friendly test, because the standard form already hands you r^2 .
Worked: the circle (x − 3)^2 + (y + 2)^2 = 16, several points
Center C = (3, −2) , r = 4 , r^2 = 16 . This is the completed-square form of x^2 + y^2 − 6x + 4y − 3 = 0 .
Point A (6, 2): (6 − 3)^2 + (2 − (−2))^2 = 9 + 16 = 25 . 25 > 16 , so A is outside . Distance is 5, which is greater than 4.
Point B (3, 2): (3 − 3)^2 + (2 + 2)^2 = 0 + 16 = 16 . 16 = 16 , so B is on the circle. Distance 4.
Point C (4, −2): (4 − 3)^2 + (−2 + 2)^2 = 1 + 0 = 1 . 1 < 16 , so C is inside . Distance 1.
Point D (3, −2): distance 0, the center itself, inside (strictly inside for r > 0).
A stem may ask which listed point lies on the circle, or which is farthest from the center. Compute d^2 for each; the largest d^2 is farthest. Here A is farthest of the four, then B on the circle, then C, then the center.
Worked: a point given before the equation is in standard form
Is (1, 1) inside the circle x^2 + y^2 − 6x + 4y − 3 = 0 ?
You already completed the square: center (3, −2) , r^2 = 16 .
(1 − 3)^2 + (1 + 2)^2 = 4 + 9 = 13 . 13 < 16 , so inside .
You can also plug (1, 1) into the general form: 1 + 1 − 6 + 4 − 3 = −3 . Interpreting the sign of the general form is messier than completing the square first; on AAF, convert to standard form, then compare d^2 to r^2 .
Expanding standard form (the reverse)
Sometimes options are in general form and the stem gives center and radius.
Center (3, −2) , radius 4:
- (x − 3)^2 + (y + 2)^2 = 16
- x^2 − 6x + 9 + y^2 + 4y + 4 = 16
- x^2 + y^2 − 6x + 4y + 13 = 16
- x^2 + y^2 − 6x + 4y − 3 = 0
That is the original equation. Expanding is how you match an option written as x^2 + y^2 + Dx + Ey + F = 0 . Completing the square is how you go the other way. If two options differ only in the constant, they came from adding the completed-square numbers to the wrong side.
AAF circle checklist
Standard form (x − h)^2 + (y − k)^2 = r^2 : center (h, k) , radius √(right-hand side) .
(x + a)^2 means h = −a . (y + b)^2 means k = −b .
Complete the square on x and on y. Add the same constants to the right side.
Radius is the square root of r^2 , never r^2 itself, and never negative.
Classify a point by comparing (x − h)^2 + (y − k)^2 to r^2 .
Diameter endpoints: center is the midpoint; r^2 is the squared distance from the center to an endpoint.
Circle equations close Algebra 2 geometry. Volume, intersecting lines, similarity/congruence, and this standard-form skill are the entire Table 11 Algebra 2 geometry list — still only 1–2 items on the CAT, so each formula has to be automatic. Mixed FREE practice: /practice/accuplacer-advanced-algebra .